$\dfrac{ 8c + 6d }{ 4 } = \dfrac{ -9c - 10e }{ -10 }$ Solve for $c$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 8c + 6d }{ {4} } = \dfrac{ -9c - 10e }{ -10 }$ ${4} \cdot \dfrac{ 8c + 6d }{ {4} } = {4} \cdot \dfrac{ -9c - 10e }{ -10 }$ $8c + 6d = {4} \cdot \dfrac { -9c - 10e }{ -10 }$ Multiply both sides by the right denominator. $8c + 6d = 4 \cdot \dfrac{ -9c - 10e }{ -{10} }$ $-{10} \cdot \left( 8c + 6d \right) = -{10} \cdot 4 \cdot \dfrac{ -9c - 10e }{ -{10} }$ $-{10} \cdot \left( 8c + 6d \right) = 4 \cdot \left( -9c - 10e \right)$ Distribute both sides $-{10} \cdot \left( 8c + 6d \right) = {4} \cdot \left( -9c - 10e \right)$ $-{80}c - {60}d = -{36}c - {40}e$ Combine $c$ terms on the left. $-{80c} - 60d = -{36c} - 40e$ $-{44c} - 60d = -40e$ Move the $d$ term to the right. $-44c - {60d} = -40e$ $-44c = -40e + {60d}$ Isolate $c$ by dividing both sides by its coefficient. $-{44}c = -40e + 60d$ $c = \dfrac{ -40e + 60d }{ -{44} }$ All of these terms are divisible by $4$ Divide by the common factor and swap signs so the denominator isn't negative. $c = \dfrac{ {10}e - {15}d }{ {11} }$